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If t=1 the Macdonald polynomials become the sums over W orbits, which are the monomial symmetric functions when the root system has type A. If we put t = q α and let q tend to 1 the Macdonald polynomials become Jack polynomials when the root system is of type A , and Heckman–Opdam polynomials for more general root systems.
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In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike other Bessel function transforms, such as the Hankel transform , this transform involves integrating over the index of the function rather than its ...
A ring of symmetric functions can be defined over any commutative ring R, and will be denoted Λ R; the basic case is for R = Z. The ring Λ R is in fact a graded R-algebra. There are two main constructions for it; the first one given below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979).
In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald . They include as special cases the Jacobi triple product identity , Watson's quintuple product identity , several identities found by Dyson (1972) , and a 10-fold product identity found by Winquist (1969) .
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The Macdonald function (Modified Bessel function of the II kind) (Abramowitz and Stegun, 1972, p.376) is defined by: ...
A user will input a number and the Calculator will use an algorithm to search for and calculate closed-form expressions or suitable functions that have roots near this number. Hence, the calculator is of great importance for those working in numerical areas of experimental mathematics. The ISC contains 54 million mathematical constants.